Fandom Rivalry: If it still had one, with the Universal Studios Dark Universe. Not only on account of being shared universes centered around monsters, but the fact Universal tried to block King Kong from appearing in a shared world with Godzilla on account of dislike for the Big G and the citation of "shared monster universes don't work". The latter justification would ring hollow given how Universal was also making its own shared Universe built on that exact premise. With the reports that the Dark Universe won't be happening anymore in large part because of the critical and financial failure of The Mummy, MonsterVerse fans see it as Laser-Guided Karma for Universal. Also because of their attempted claim to King Kong being despite their own prior insistence that he was public domain, which Nintendo successfully used against them in 1984.
Fans of the Marvel Cinematic Universe get along very well with Monsterverse fans. It helps that they share quite a few actors, and the Monsterverse movies are not direct genre competitors, unlike with the DCEU. Plus, Marvel had produced a Godzilla comic book line in the early 80's, so many consider Godzilla an honorary Marvel character. Made even better by Michael Dougherty, director of Godzilla: King of the Monsters (2019), who tweeted a comic book cover of Marvel's Godzilla featuring the Avengers and said, "This movie would make the world a better place." Fans of both the Monsterverse and the MCU took very kindly to this tweet and expressed love for the idea of a crossover between the two.
DC Extended Universe fans are on mostly good terms with Monsterverse fans. Because both are produced by Warner Bros. and because both (initially for the DCEU) have a grounded take on both film genres (Kaiju and superheroes). But it could also be the fact that Legendary Pictures did produce Man of Steel, the DCEU's first entry.
A factor that is obviously present and will be present as it goes on, but also probably a reason that impacts the above relationships as well. A portion of this franchise's audience is only really going to care about monster attacks and monster battles.
Alternatively, some people see the movies because of the human cast. In fact, some people who went to see Skull Island because the movie has big-name actors like Tom Hiddleston, Brie Larson, and Samuel L. Jackson.
Memetic Mutation: SAVE MOTHRA! Explanation A spoof of the rather polarizing Martha scene from Batman v Superman: Dawn of Justice in which Batman and Superman end their conflict once Superman pleads Batman to Save Martha!, a name both characters Mothers share. Since Martha sounds very similar to Mothra, this has led to numerous fans making jokes that Kong and Godzilla will end their upcoming fight once one of them (either Kong or Godzilla) pleads to the other to SAVE MOTHRA!. Naturally, it ends with Kong or Godzilla demanding WHY DID YOU SAY THAT NAME?
Only the Creator Does It Right: Toho-Godzilla "purists" are not as accepting of this universe's take on the character despite Toho's full endorsement of it.